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oai:drops-oai.dagstuhl.de:6327 2024-03-06T10:37:43Z ddc:004 open_access

Parity Separation: A Scientifically Proven Method for Permanent Weight Loss Curticapean, Radu perfect matchings counting complexity structural complexity exponentialtime hypothesis Given an edge-weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G. In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge-weights 1 and -1, we construct two unweighted graphs G1 and G2 such that PerfMatch(G) = PerfMatch(G1) - PerfMatch(G2). This yields a novel weight removal technique for counting perfect matchings, in addition to those known from classical #P-hardness proofs. Our technique is based upon the Holant framework and matchgates. We derive the following applications: Firstly, an alternative #P-completeness proof for counting unweighted perfect matchings. Secondly, C=P-completeness for deciding whether two given unweighted graphs have the same number of perfect matchings. To the best of our knowledge, this is the first C=P-completeness result for the “equality-testing version” of any natural counting problem that is not already #P-hard under parsimonious reductions. Thirdly, an alternative tight lower bound for counting unweighted perfect matchings under the counting exponential-time hypothesis #ETH. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Radu Curticapean 2016 Is Part Of LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2016.47 urn:nbn:de:0030-drops-63279 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.47 eng https://creativecommons.org/licenses/by/3.0/legalcode